These developments are not only inspired by mentioned Tate's thesis but also by A.Borisov's work on arithmetic cohomology theory introduced in [1]}. Even though we operate with eigenfunctions of the global Fourier transform instead of cohomology spaces, one can see that those two approaches encode the same data (see REF ). It is worth pointing out that the Relative Riemann-Roch theorem ([2]}) is applied ([2]}) in the proof of the Falting-Riemann-Roch theorem ([2]}. Therefore the theory developed here is not only interesting on its own, but also is a little step towards a two-dimensional generalization.
